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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x0</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">y0</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">y1</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">n</font>;<br/>













<b>assume </b><a NAME="E1:86"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">L1</font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">L2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x2</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">x3</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">y2</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">y3</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:86"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x2</font>,<font color="Maroon">x3</font> &amp; <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y2</font>,<font color="Maroon">y3</font> &amp; <font color="Maroon">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E8"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:86"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclidlp.html#T23" target="_self">Th23</a>, </i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:86"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclidlp.html#T23" target="_self">Th23</a>, </i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:86"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T23" target="_self">Th23</a>, </i>;<br/>
<a NAME="E10:86"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, </i>;<br/>
<a NAME="E11:86"/><b>then </b><i><font color="Green">E106</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T25" target="_self">Th25</a>, <a class="ref" href="euclidlp.html#T26" target="_self">Th26</a>, <a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>;<br/>
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a>, <a class="ref" href="euclidlp.html#T25" target="_self">Th25</a>, <a class="ref" href="euclidlp.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T29" target="_self">Th29</a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T26" target="_self">Th26</a>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/>
<b>hence </b><a NAME="E13:86"/>
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T28" target="_self">Th28</a>, </i>;<br/>


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